Our first approach (sphere \((a)\)  in Figure REF ) leverages the fact that the constraint function is reasonably simple and \(r_{n,\mathcal {M}}\)  can thus be integrated numerically and with high precision. Thus, the CGFD is calculated explicitly using this approximation of the normalizing constant. The second approach (sphere \((b)\) ) reparameterizes the problem using \(\mu := ( \cos \theta \sin \phi , ~\sin \theta \sin \phi , ~\cos \phi )^{\prime }\)  for \(0 \leqslant \theta < 2 \pi ,\)  \(0 < \phi < \pi .\)  Again, due to the relative simplicity of this example, numeric integration is available to calculate the normalizing constant of the GFD from Theorem REF  explicitly. For the last two approaches (spheres \((c)\)  & \((d)\) ), we simulate draws from the CGFD using the algorithms in Sections REF  & REF , using Theorem REF  to derive an expression for the derivative in the [1]} algorithm. For the Brubaker algorithm, we draw \(\sim 2e4\)  samples, and for the Zappa algorithm we draw a larger \(2e6\) . The difference in number of samples was due to the fact that the Zappa algorithm was slower to converge, yet each individual MCMC step was quicker to calculate. For each approach, half of the samples were thrown out as burn-in. Comparing these four approaches investigates the effectiveness of the simulation algorithms while also verifying that the CGFD is invariant to different parameterizations (Theorem REF ).
